Abstract
In a sequence of recent results (PODC 2015 and PODC 2016), the running time of the fastest algorithm for the minimum spanning tree (MST) problem in the Congested Clique model was first improved to O(log(log(log(n)))) from O(log(log(n))) (Hegeman et al., PODC 2015) and then to O(log^*(n)) (Ghaffari and Parter, PODC 2016). All of these algorithms use Theta(n^2) messages independent of the number of edges in the input graph.
This paper positively answers a question raised in Hegeman et al., and presents the first "superfast" MST algorithm with o(m) message complexity for input graphs with m edges. Specifically, we present an algorithm running in O(log^*(n)) rounds, with message complexity ~O(sqrt{m * n}) and then build on this algorithm to derive a family of algorithms, containing for any epsilon, 0 < epsilon <= 1, an algorithm running in O(log^*(n)/epsilon) rounds, using ~O(n^{1 + epsilon}/epsilon) messages. Setting epsilon = log(log(n))/log(n) leads to the first sublogarithmic round Congested Clique MST algorithm that uses only ~O(n) messages.
Our primary tools in achieving these results are
(i) a componentwise bound on the number of candidates for MST edges, extending the sampling lemma of Karger, Klein, and Tarjan (Karger, Klein, and Tarjan, JACM 1995) and
(ii) Theta(log(n))wiseindependent linear graph sketches (Cormode and Firmani, Dist. Par. Databases, 2014) for generating MST candidate edges.
BibTeX  Entry
@InProceedings{pemmaraju_et_al:LIPIcs:2016:6882,
author = {Sriram V. Pemmaraju and Vivek B. Sardeshmukh},
title = {{SuperFast MST Algorithms in the Congested Clique Using o(m) Messages}},
booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
pages = {47:147:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770279},
ISSN = {18688969},
year = {2016},
volume = {65},
editor = {Akash Lal and S. Akshay and Saket Saurabh and Sandeep Sen},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6882},
URN = {urn:nbn:de:0030drops68827},
doi = {10.4230/LIPIcs.FSTTCS.2016.47},
annote = {Keywords: Congested Clique, Minimum Spanning Tree, Linear Graph Sketches, Message Complexity, Sampling}
}
Keywords: 

Congested Clique, Minimum Spanning Tree, Linear Graph Sketches, Message Complexity, Sampling 
Collection: 

36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016) 
Issue Date: 

2016 
Date of publication: 

10.12.2016 